The video is indeed rather silly, not in what it says but what it does not say. However, in its proper mathematical context the answer is completely correct and probably the best way to understand is to take a look at the classic text on this topic: G.H. Hardy CDivergent Series, published in 1947. This result and many others (such as 1+1+1+ ... = -1/2) is at the end of section 13. Of course, (as psycho_dad has already pointed out) in Mathematica this is just Zeta[-1] and 1+1+1+ ... -1/2 is simply Zeta[0]. But probably a more convincing argument is by using Ramanujan's summation, which is explained in Hardy's book in detail.

> You may try the Regularization option for Sum, but it doesn't seem to give any finite result for that divergent series.>> On the other hand, the video to which you refer relies ultimately upon using Ces=E0ro-summability of 1 - 1 + 1 - 1 _ . . . , which you may implement in Mathematica as:>> Sum[(-1)^n, {n, 0, \[Infinity]}, Regularization -> =93Cesaro"]> (* 1/2 *)>> [The video to which you refer is disingenuous in not saying up-front that it's not using ordinary summability but some other form(s) of summability. (The merest hint is a brief glimpse of a page of a text on String Theory where the formula> 1 + 2 + 3 + . . . = -1/12 is displayed just below a line referring to renormalization.)>> As it stands, that video, in my mind, is deleterious to understanding of the mathematics of infinite series destructive of trust in mathematics: it manipulates divergent series as if they were convergent.]>>> On Jan 19, 2014, at 2:56 AM, Matthias Bode <lvsaba@hotmail.com> wrote:>>>>> Hola,>>>> I came across this video (supported by the Mathematical Sciences Research Institute* in Berkeley, California):>>>> http://www.numberphile.com/videos/analytical_continuation1.html>>>> Could the method shown in this video be replicated using Mathematica symbols such as Sum[] &c.?>>>> Best regards,>>>> MATTHIAS BODES 17.36398=B0, W 66.21816=B0,2'590 m. AMSL.>>>> *) http://www.msri.org/web/msri>>>> Murray Eisenberg murray@math.umass.edu> Mathematics & Statistics Dept. > Lederle Graduate Research Tower phone 240 246-7240 (H)> University of Massachusetts > 710 North Pleasant Street > Amherst, MA 01003-9305>>>>>>>